Computing elliptic integrals by duplication

Abstract

Summary. Logarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the convergence is improved by including a fixed number of terms of Taylor's series, the error ultimately decreases by a factor of 4096 in each cycle of iteration. Except for Cauchy principal values there is no separation of cases according to the values of the variables, and no serious cancellations occur if the variables are real and nonnegative. Only rational operations and square roots are required. An appendix contains a recurrence relation and two new representations (in terms of elementary symmetric functions and power sums) for R-polynomials, as well as an upper bound for the error made in truncating the Taylor series of an R-function.

Inversion of Physical Properties for Determining the Microstructure of Natural Composites

“…We used Carlson's algorithm (Carlson, 1979) to calculate the elliptical integrals F(θ,k) and E(θ,k). The components of the Wu's tensor T (i) (Wu, 1966) were calculated numerically using equations (4).…”

Computing Toroidal Functions for Wide Ranges of the Parameters

“…The CF from Eq. (4) is more rapidly convergent as x becomes larger, while the convergence of the CF in Eq. (5) is faster as x becomes smaller (but x > 1 ).…”

“…Obtain the starting values, Q~1 12 and Q~1 12 through their relation with elliptic integrals (evaluated by using Carlson's duplication theorem [4,5] …”

Magnetohydrodynamic implosion symmetry and suppression of Richtmyer-Meshkov instability in an octahedrally symmetric field

“…For example, if a current loop is centered on the x axis (as is the case for this field configuration), then q = x. E and K are respective elliptic integrals of the first and second kind (calculated here according to the algorithm of Carlson [24]), and r ra is the projected radial position for the ith current loop. α i = {α 0 , − α 0 } scales the magnetic field such that the magnetic field strength parameter β 0I = 2p 0 /|B(R 0 )| 2 , and Table I for description of case abbreviations).…”